On peak sets for the real part of a function space (Q1061980)

Let A be a function space on a compact Hausdorff space X. The authors give conditions for certain families of closed subsets in X under which A are characterized. A closed subset F in X is called an NPEP-set for A if for any \(f\in A\) there is a \(g\in A\) such that \(g=f\) on F and \(\| g\| =\| f\|_ F\). \textit{E. Briem} showed that \(A=C(X)\) if any closed subset F in X is an NPEP-set for A [Math. Z. 178, 421-427 (1981; Zbl 0449.46019)]. The authors first obtain a slight extension of the theorem of Briem above, and by using the argument in the result give a theorem on peak sets for ReA in the case of function spaces in association with a theorem of \textit{E. Briem} [Proc. Am. Math. Soc. 85, 77-78 (1982; Zbl 0487.46032)] for peak sets for ReA in the case where A are function algebras.